FINDING N-TH ROOTS IN NILPOTENT GROUPS AND ...

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576 S. Sze, D. Kahrobaei, R. Dambreville, M. Dupas Definition 3.2. A subgroup A of an R−group G is said to be an isolated subgroup if for every element a ∈ A and every n ∈ N the solution to x n = a, if it exists, is an element in A. Definition 3.3. Let M be a set of elements of an R−group G. The isolated closure, or isolator of M in G, denoted I(M), is the unique minimal isolated subgroup of G containing M. A normal subgroup N of an R−group G is isolated if and only if the factor group G/N is torsion-free. Proposition 3.4. If H is an isolated subgroup of an R−group G and if G/H is an R−group (note that this isn’t always true), then in the natural one-to-one correspondence between all subgroups of G/H and all subgroups of G containing H, isolated subgroups correspond to each other. In other words, isolated subgroups of G/H correspond to isolated subgroups of G containing H. Proof. Suppose that A/H is an isolated subgroup of G/H and we have g n = a for g ∈ G and a ∈ A. We want to show that g ∈ A. Since g n = a and A/H is an isolated subgroup of G/H, then (gH) n = aH implies that gH ∈ A/H so that g ∈ A. Now suppose that A is an isolated subgroup of G containing H and that (gH) n = aH for g ∈ G and a ∈ A. This implies that g n = ah ∈ A. Since A is isolated, this means that g ∈ A so gH is an element of A/H, hence isolated. Proposition 3.5. The centralizer of an arbitrary set of elements of an R−group G is an isolated subgroup. Proof. Let M be an arbitrary set of elements of G. Let x an element of G such that x n is an element of the centralizer of M. Then a −1 x n a = x n for all a ∈ M implies that (a −1 xa) n = x n . Since G is an R−group, this means that a −1 xa = x, so x ∈ CG(M) as well. It follows that the center of an R−group is isolated. Proposition 3.6. In every R−group G, the equation a k b ℓ = b ℓ a k , where a, b ∈ G, implies that ab = ba. Proof. It suffices to prove this proposition when one of k or ℓ is 1. Without loss of generality, we assume that ℓ = 1 so that we have a k b = ba k and we want to show that this implies ab = ba for a and b in an R−group G. Now, (b −1 ab) k = b −1 a k b = b −1 ba k = a k . Since G is an R−group, this implies that b −1 ab = a.
FINDING N-TH ROOTS IN NILPOTENT GROUPS... 577 Proposition 3.7. A torsion-free group is an R−group if and only if the factor group of its center is an R−group. Proof. Let G be a torsion-free R−group. Since the center Z = Z(G) of G is isolated, we know G/Z is torsion-free. We want to show that G/Z is an R−group. Suppose (xZ) n = (yZ) n , for x, y ∈ G. Therefore, x n = y n z for z ∈ Z. This means that y −n x n = z. We also have x n = zy n so x n y −n = z. Hence, x n y −n = y −n x n . From Proposition 3.6, this means that xy = yx. Now, x n = y n z and xy = yx implies that (y −1 x) n = z. Since Z is isolated, y −1 x ∈ Z, so xZ = yZ. Conversely, let G be torsion-free and let the factor group G/Z be an R−group. Suppose that x n = y n for x, y ∈ G. Then (xZ) n = (yZ) n . Since G/Z is an R−group, this implies that xZ = yZ, or x = yz, z ∈ Z. Raising this to the n-th power gives x n = (yz) n = y n z n . But the assumption that x n = y n now tells us that z n = 1. Since G is torsion-free, this implies that z = 1, or x = y. Proposition 3.8. All the terms of the upper central chain {1} = Z0 Z1 Z2 · · · Zα ... of an R−group G are isolated in G, all the factors Zα/Zα−1 of this chain are abelian torsion-free groups, and all the factor groups G/Zα are R−groups. Proof. Since Z1 is the center of G, it is isolated in G. The factor group Z1/Z0 = Z1 is an abelian torsion-free group and the factor group G/Z1 is an R−group. Suppose that the theorem has been proved for α < β. If β − 1 exists, then G/Zβ−1 is an R−group and its center, Zβ/Zβ−1, is an abelian torsion-free group; the factor group of the center, isomorphic to G/Zβ, is an R−group (by Proposition 3.7); and from the fact that the center of the R−group G/Zβ−1, Zβ/Zβ−1, is isolated, we also have Zβ is isolated (by 3.4). If β is a limit number, then Zβ is the union of an ascending sequence of isolated subgroups and is therefore isolated. Furthermore, if (xZβ) n = (yZβ) n for x, y ∈ G, then x n = y n z for z ∈ Zβ, and therefore z ∈ Zα, α < β. Therefore, (xZα) n = (yZα) n and since G/Zα is assumed to be isolated, this implies that xZα = yZα. From this, we conclude that xZβ = yZβ so G/Zβ is an R−group. Proposition 3.9. The isolator of an arbitrary element other than the unit element of an R−group is an abelian torsion-free group of rank 1 and is the isolator of each of its elements other than 1. The isolators of any two elements of an R−group either are identical or else intersect in the unit element.
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